What is the set of all functions, and as a consequence the set of all m x n matrices
supposed to look like?

S = {(x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R} is a regular vector space, another notation I've seen for this is:

S = {α ∊ (x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R} which clearly indicates α is a vector and the
operations defined on S are:

+ : S x S → S defined by + : (α,β) ↦ α + β = (x₁,x₂,...,x₊) + (y₁,y₂,...,y₊) = ...

: S x F → F defined by : (α,c) ↦ (cα) = c(x₁,x₂,...,x₊) = ...

Now, if we look at the set of functions we see that:

+ : S x S → S defined by + : (f,g) ↦ (f + g)(x) = f(x) + g(x)
: S x F → F defined by : (f,β) ↦ (βf)(x) = βf(x).

so would my set be

S = {(f,g,...,j)|f,g,...,j∊R}


S = {f ∊ (x₁,x₂,...,x₊)|x₁,x₂,...,x₊∊R}?

I can't get the set of functions to follow the format I've used above
I don't think either of those sets I've just described make sense either tbh

Then with matrices, if we look at how + & are defined for functions and notice
that a matrix is just the function
f : (i,j) ↦ A(i,j) = Aij
it seems reasonable that the vector space of matrices is defined in the same way
as the vector space of functions.

I am really not sure, I have a feeling that for matrices the operations are
defined along the lines of:

+ : $\displaystyle F^{m x n}$ x $\displaystyle F^{m x n}$ → $\displaystyle F^{m x n}$ defined by + : (i,j) ↦ (A + B)(i,j) = A(i,j) + B(i,j) = Aij + Bij

: $\displaystyle F^{m x n}$ x F →$\displaystyle F^{m x n}$ defined by : ((i,j),β) ↦ (βA)(i,j) = βA(i,j) = βAij

rather than

+ : S x S → S defined by + : (f,g) ↦ (f + g)(x) = f(x) + g(x)
: S x F → F defined by : (f,β) ↦ (βf)(x) = βf(x)

because that notation suggests the m x n dimension characteristic of matrices
where the standard function notation obscures it (I think). Notice the last bit of
notation (the (f,g) ↦ (f + g)(x) stuff for matrices) really doesn't make sense either
so I don't think it can be along these lines.

So, to sum up I'm just asking about the notation describing the vector space of
functions and as a consequence of this the notation for the vector space of all
mxn matrices.

What say you?